I’d start by recommending a timestamp scheme so you can easily find the statement you’re looking for in the dropdown (e.g. yymmdd-bank = “200709-WellsFargo”).
There shouldn’t be an issue with deleting rows from the Statement sheet. You may find that those cells have broken data validation in the Transactions sheet (i.e. the reference statement name no longer exists in the Statements sheet). I’d delete one Statement row and see if that causes issues in your spreadsheet before deleting many.
I’m not sure what you mean about showing “only the most current Statement in the Transaction sheet”… Do you mean in the dropdown? That is not possible (all statement names from the Statement sheet will show). If you mean you want to filter the Transactions sheet, you can use the Google Sheets filter to filter on the Statement column. Alternatively, you can install the Statement Details sheet using Tiller Money Labs.